Today I came across an assertion that Plato used as an approximation of . Indeed, it is not a bad approximation: (although it is not within Archimedes’s bounds: ).
Not only had I not seen this approximation before, I had not heard of any value of attributed to Plato.
I investigated a little further and discovered that there is no direct evidence that Plato knew of this approximation. It was pure speculation by the famous philosopher of science Karl Popper! Here’s what Popper has to say (this is in his notes to Chapter 6 of The Open Society and its Enemies, Vol. 1, pp. 252–253).
It is a curious fact that very nearly approximates … The excess is less than , i.e. less than pro mille of , and we have reason to believe that no better upper boundary for had been proved to exist. A kind of explanation of this curious fact is that it follows from the fact that the arithmetical mean of the areas of the circumscribed hexagon and the inscribed octagon is a good approximation of the area of the circle. Now it appears, on the one hand, that Bryson operated with the means of circumscribed and inscribed polygons,… and we know, on the other hand (from the Greater Hippias), that Plato was interested in the adding of irrationals, so that he must have added . There are thus two ways by which Plato may have found out the approximate equation , and the second of these ways seems almost inescapable. It seems a plausible hypothesis that Plato knew of this equation, but was unable to prove whether or not it was a strict equality or only an approximation.
Popper then spends a couple of paragraphs tying this into an earlier discussion of Plato’s Timaeus. This is the work in which Plato discusses the four elements (earth, air, fire, and water) and associates them with four of the regular polyhedra (cube, octahedron, tetrahedron, and icosahedron, respectively). The connection between Timaeus and is the relation between the values and , the 45-45-90 and 30-60-90 triangles which can be used to make the faces of the polyhedra, and the approximations to the area of a circle using these triangles.
Popper ends with the reminder/disclaimer:
I must again emphasize that no direct evidence is known to me to show that this was in Plato’s mind; but if we consider the indirect evidence here marshalled, then the hypothesis does perhaps not seem too far-fetched.
Note: if we take the unit circle and construct a circumscribed hexagon and an inscribed octagon, then the area of the hexagon is and the area of the octagon is . So, it is true that the average of these areas is .